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In mathematics, a limit point of a set ''S'' in a topological space ''X'' is a point ''x'' (which is in ''X'', but not necessarily in ''S'') that can be "approximated" by points of ''S'' in the sense that every neighbourhood of ''x'' with respect to the topology on ''X'' also contains a point of ''S'' other than ''x'' itself. Note that x does not have to be an element of S. This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set and topological closure. Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by uniting it with its limit points. ==Definition== Let ''S'' be a subset of a topological space ''X''. A point ''x'' in ''X'' is a limit point of ''S'' if every neighbourhood of ''x'' contains at least one point of ''S'' different from ''x'' itself. Note that it doesn't make a difference if we restrict the condition to open neighbourhoods only. This is equivalent, in a ''T''1 space, to requiring that every neighbourhood of ''x'' contains infinitely many points of ''S''. It is often convenient to use the "open neighbourhood" form of the definition to show that a point is a limit point and to use the "general neighbourhood" form of the definition to derive facts from a known limit point. Alternatively, if the space ''X'' is sequential, we may say that ''x'' ∈ ''X'' is a limit point of ''S'' if and only if there is an ω-sequence of points in ''S'' \ whose limit is ''x''; hence, ''x'' is called a ''limit point''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Limit point」の詳細全文を読む スポンサード リンク
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